\section{Background}
\label{sec:Background}

PCIR advocates assume that health care providers are wasteful and inefficient, providing more services than needed to generate more income. PCIR advocates believe that fee for service providers over-test, over-diagnose, and over-treat patients and assume that capitation encourages providers to become less inefficient. PCIR critics fear that risk assuming providers will under-test, under-diagnose, and under-treat patients \citep{BBlackwell08011988, Blomqvist2005, DouglasAConrad09012004, ALHillman07131989, AllisonKempe05012000, Jonas2003}.

Unethical, profit maximizing, providers exist in both capitated and fee for service payment systems, maximizing net income by delivering excessive care under fee for service and inadequate care under capitation. These unethical providers harm patients by exposing them to excessive testing and intervention or by failing to diagnose and treat them as early as they could.

There is also widespread misunderstanding about insurance. \cite{Jonas2003} (See footnote pg 9) suggests that health insurance is not ``real insurance'' because people will use all their benefits eventually. Correcting misconceptions about insurance and PCIR is difficult because few people recognize that the most important consideration is insurer size. With regard to Jonas' concern, we note that ``whole life'' insurance always pays full benefits so it is not whether the payment of full benefits is feasible, but the timing of premiums and benefits that determine whether any insurance system is sustainable and feasible \citep{Bowers:2e:1997, Buhlmann:83, GVK017764068, Borch, Kellison}.

Risk managers, health policy analysts and providers must understand insurance operations, rate making and financial reporting. This paper will contribute to the development of such capabilities. We begin by avoiding faulty assumptions about waste and inefficiency that hide the inefficiency of the insurance risk transfers that have been the mainstay of cost control efforts for the last 4 decades, concentrating on the effect insurer size has on insurer's operations and operating results when health care finance systems are efficient. 	

I can describe the flaws in capitation with a simple model and familiar statistical tools. By assuming that the health care (finance) systems are operating as efficiently as possible, I can show that when capitation is implemented in efficient health care (finance) systems, differences in portfolio size, and the differences in risk management efficiency that result, make it impossible for efficient health care providers, or any small insurers, to offer benefits at the pre-transfer level under the influence of capitation.

The key to insurance is the different degrees of variation in insurers' loss ratios as functions of insurer portfolio size \citep{Bowers:2e:1997, Borch, Buhlmann:80, Gerber1979, Salzmann1984}. Many financial analysts,  risk managers, and health care providers assume that both large and small insurers have identical operating outcomes. Although large and small insurers do have identical ``expected loss ratios'' when they select policyholders at random, from identical populations, they do not have identical loss ratio probability distributions so they do not have identical probabilities of financial outcomes that are dependent on their actual, rather than their expected loss ratios. 

Starting with modest, market appropriate assumptions about a large ``Paradigm Insurer'' (PI), I explain how the Central Limit Theorem applies to health insurance and capitation, by analyzing the impact of portfolio size on insurer operating characteristics \citep{Fel68}. I cannot account for all the risk insurers' face. Instead, I focus on the routine variation in insurer's fortunes. A pandemic would sap the strength of any health insurer, no matter how well capitalized it may be, and most health insurers, managed care organizations, and risk assuming health care providers would not survive even a fairly modest epidemic. The analyses below focus on insurer's non-cataclysmic risk exposure, the most common financial risks faced by risk assuming health care providers.
